SetPartitionDimension on SetPartition
The sum of the lengths of the arcs, taking for each block its largest element minus its smallest.
FindStat's St000728; the arcs join consecutive elements of a block, so the lengths add up to the block's span.
Called as CombinatorialStat(x, "SetPartitionDimension"). The head's page has its examples.
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Distribution
Over SetPartitions at each size, tallied here in your browser; sizes past 10 are not offered.
FindStat St000728
The dimension of a set partition.
| Object | Value |
|---|---|
{{1,2}} | 1 |
{{1,2,3}} | 2 |
{{1},{2}} | 0 |
{{1,2,3,4}} | 3 |
{{1,2},{3}} | 1 |
{{1,3},{2}} | 2 |
{{1},{2,3}} | 1 |
{{1,2,3,4,5}} | 4 |
Data from FindStat, the combinatorial statistic finder; the page at findstat.org/St000728 has the rest.